This easily splits into the two disjoint cases where \(p \nmid n\)  or \(n=p^a\) . When \(p \nmid n\) , the result follows from Huber's much more general base change results [1]}. We may thus assume that \(n=p^a\) . By Theorem REF , we know that \(g^\ast Rf_\ast {F}\)  and \(Rf^{\prime }_\ast g^{\prime \ast }{F}\)  are Zariski-constructible, hence overconvergent, so it suffices to show that the base change map induces an isomorphism on stalks at all rank one geometric points \(\overline{y} \rightarrow Y^{\prime }\) . By two applications of [1]}, we compute that \((g^\ast Rf_\ast {F})_{\overline{y}} \simeq R\Gamma (X \times _{Y} \overline{g(y)},{F})\)  and \((Rf^{\prime }_\ast g^{\prime \ast }{F})_{\overline{y}} \simeq R\Gamma (X \times _{Y} \overline{y},{F})\) . We now conclude by Lemma REF .
